# Question about identity of Dirac delta function [duplicate]

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I am trying to understand an identity of the $\delta$-function written on this Wikipedia page: \begin{equation} \int \mathrm{d} x \; f(x) \delta[g(x)] = \sum\limits_i \frac{f(x_i)}{\left| \frac{dg(x_i)}{dx}\right|} \tag{1} \end{equation} where $x_i$ are the zeros of $g(x)$ (i.e. $g(x_i)=0$). Now, my question is about the denominator on the right-hand side of equation $(1)$. Is that supposed to be a Jacobian determinant: \begin{equation} \left|\frac{dg(x_i)}{dx}\right| \overset{?}{=} \mathrm{det}\left(\frac{dg(x_i)}{dx}\right) \end{equation} or is it supposed to mean the modulus? The reason I think it might be a Jacobian is because Wiki mentions that we can use the following identity to change variables of integration: \begin{equation} \int_{\mathbf{R}} \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) |g'(x)|\,dx = \int_{g(\mathbf{R})} \delta(u)f(u)\, du \end{equation}

## marked as duplicate by Chris Janjigian, vonbrand, LutzL, TMM, M TurgeonMar 11 '14 at 1:07

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• And has an answer there that is much more explicit than my answer. – LutzL Mar 11 '14 at 0:33

## 1 Answer

The first variant. Everything is in one real variable, so you do not get Jacobian matrices to compute determinants.

The best way to understand that identity is to think of a delta-approximating sequence with compact support, for instance based on the quadratic or cubic B-Spline. Then consider small disjoint intervals around the roots of $g$, make the index in the delta-approximation so large that the support of the approximation is inside the images of these intervals, and perform standard parameter substitution on each of the intervals.

• Thanks for your answer. Just to be clear, if the integral was over multiple variables, and the functions $f$ and $g$ depended on those multiple variables, then we do compute the Jacobian matrices? – Hunter Mar 11 '14 at 1:27
• Then the same applies, only that instead of intervals you take balls around the roots where $g'(x)$ is invertible. And then indeed it is the absolute value of the determinant in the denominator. – LutzL Mar 11 '14 at 1:45